Connected planar graph Euler edge bound
ProvedWilliamsonShmoys.connected_planar_graph_edge_bound__p2m_eac4b030_cd26_487c_98af_ec9d769936c5_a1approximation-algorithmsplanar-graphstreewidth
A finite simple graph on at least three vertices with a connected crossing-free drawing has at most 3|V|−6 edges. This removes the topological and Euler-counting obligation from the parent; the parent applies it to H. Its exact connectedness and size hypotheses are preserved. A proof route is to derive a rotation system from the drawing, apply Euler’s formula and incidence counting, and bound face lengths using simplicity and the three-vertex condition.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn set_option autoImplicit false open WilliamsonShmoys
Formal statement
theorem WilliamsonShmoys.connected_planar_graph_edge_bound__p2m_eac4b030_cd26_487c_98af_ec9d769936c5_a1
{V : Type} [Fintype V] [LinearOrder V]
(H : SimpleGraph V)
(hHconn : ∀ u v, Relation.ReflTransGen H.Adj u v)
(hHdraw : WilliamsonShmoys.HasPlanarDrawingOn H)
(hV : 3 ≤ Fintype.card V) :
H.edgeSet.ncard + 6 ≤ 3 * Fintype.card V := by sorrySource
David P. Williamson and David B. Shmoys, The Design of Approximation Algorithms, Cambridge University Press, 2011, author electronic manuscript, Section 10.2, proof of Theorem 10.11, pp. 270-272. https://doi.org/10.1017/CBO9780511921735. Bounded tree-width of the pieces: Robertson–Seymour (planar graphs of radius r have tree-width at most 3r+1) / Bodlaender (k-outerplanar graphs have tree-width at most 3k-1).